Chebyshev polynomial approximation to approximate partial differential equations

Least-Squares algorithm introduced in Longstaff and Scwartz (2001). The coefficient
c in Section 3 are still estimated using Chebyshev polynomials and the same approach as
before but now the interpolant nodes are projected using (7)⁸. We assume that there is a
threshold at $0.87 and once it has been reached one has to decide if it is convenient to
shutdown the machinery. Using 200 time steps, four Chebyshev basis functions and one
hundred thousands replications, we estimate that it is optimal to shutdown the machinery
soon after its second year of life. This empirical result is in line with the three paths
shown in Figure (3).

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We have aslo tried to estimates the parameters using Chebyshev nodes but in this specifi case we obtained apoor fit. Therefore we suggest, when dealing with
optimal stopping problems, to not use Chebyshev nodes.

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